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Lasers: Population Inversion, Gain, Cavities, Rate Equations

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Lasers: Population Inversion, Gain, Cavities, Rate Equations What Makes a Laser • Light Amplification by Stimulated Emission of Radiation Main Requirements of the Laser • Laser Gain Medium (provides the light amplification) • Optical Resonator Cavity (greatly increase amplification) Also creates small beam • Sufficient means of Excitation of Gain medium (called pumping)eg. light, current, chemical reaction • Population Inversion in the Gain Medium due to pumping Creates the stimulated emission of radiation Laser Types • Two main types depending on time operation • Continuous Wave (CW) • Pulsed operation • Pulsed is easier, CW more useful Regular Light Sources: Equilibrium Energy Populations • Laser are quantum devices • Assume gas in thermal equilibrium at temperature T • Some atoms in a Gas are in an excited state • Quantization means discrete energy levels 3 • Atoms Ni (atoms/m ) at a given energy level Ei • E0 is the ground state (unexcited) • Fraction at a given energy follows a Boltzmann distribution N ⎛ [E − E ]⎞ i = exp⎜− i 0 ⎟ N0 ⎝ KT ⎠ T = degrees K K = Boltzman constant 1.38 x 10-23 J/K = 8.62 x 10-5 eV/K Spontaneous and Stimulated Emission • Consider 2 energy levels E0 (ground state) and E1 (excited state) • Photon can cause Stimulated Absorption E0 to E1 • Excited state has some finite lifetime, τ10 (average time to change from state 1 to state 0) • Spontaneous Emission of photon when transition occurs • Randomly emitted photons when change back to level 0 • Passing photon of same λ can cause "Stimulated Emission" • Stimulated photon is emitted in phase with causal photon • Stimulated emission the foundation of laser operation Einstein's Rate Equations • Between energy levels 2 and 1 the rate of change from 2 to 1 is dN 21 = −A N dt 21 2 -1 • where A21 is the Einstein Coefficient (s ) • After long time energy follows a Boltzmann distribution N ⎛ [E − E ]⎞ 2 = exp⎜− 2 1 ⎟ N1 ⎝ KT ⎠ • If (E2 - E1) >> KT then over a long time N2( t ) = N2( 0 ) exp(A21t) • Thus in terms of the lifetime of the level τ21 sec, 1 A21 = τ 21 • illuminated by light of energy density ρ = nhν (J/m3) 3 (n= number of photons/m ) of frequency ν12 the absorption is • At frequency ν12 the absorption is dN 1 = N B ρ()ν emissions dt 1 12 12 m3s • B12 is the Einstein absorption coefficient (from 1 to 2) • Similarly stimulated emission rate (with B21=B12) is dN 2 = N B ρ()ν emissions dt 2 21 21 m3s Two level system: Population Inversion • In thermal equilibrium lower level always greater population • N1 >> N2 • Can suddenly inject energy into system - pumping • Now not a equilibrium condition • If pumped hard enough get "Population Inversion" • Upper level greater than lower level: N2 >> N1 • Population Inversion is the foundation of laser operation Creates the condition for high stimulated emission • In practice difficult to get 2 level population inversion • Best pumping with light gives is equal levels • Reason is Einstein’s rate equations dN dN 2 = N B ρ()ν = N B ρ ()ν = 1 emissions dt 2 21 21 1 12 21 dt m3s • Since B21=B12 then N1=N2 with light pumping • Need more levels to get population inversion Three level systems • Pump to E0 level E2, but require E2 to have short lifetime • Rapid decay to E1 • E1 must have very long lifetime: called Metastable • Now population inversion readily obtained with enough pumping • Always small amount of spontaneous emission (E1 to E0) • Spontaneous create additional stimulated emission to E0 • If population inversion: stimulated emission dominates: Lasing • Common example Nd:Yag laser • Problem: E0 often very full He-Ne Laser Energy levels • He-Ne laser is example of 3 level laser • Levels are created in both He and Ne atoms • Fast decay from lower state (2p) to ground state • First laser (Ruby) was a 3 level system Four Level Systems • Pump to level E3, but require E3 to have short lifetime • Rapid decay to E2 so E3 is always empty • E2 must have very long lifetime: metastable so fills up • Also require E1 short lifetime for decay to E0 • Now always have E1 empty relative to E2 • In principal easier to get population inversion • Always small amount of spontaneous emission (E2 to E1) • Spontaneous photons create additional stimulated emission to E1 • If population inversion: stimulated emission dominates • Problem: low efficiency: energy losses at E3 to E2 and E1 to E0 Absorption in Homogeneous Mediums • Monochromatic beam passing through absorbing medium homogeneous medium • Change in light intensity I is ΔI = I(x + Δx)− I(x) ΔI = −αΔxI(x) where α = the absorption coefficient (cm-1) • In differential form dI(x) = −αI(x) dx • This differential equation solves as I(x) = I0 exp(−αx) Gain in Homogeneous Mediums • If we have a population inversion increase I • Stimulated emission adds to light: gain I(x) = I0 exp(gx) g = small signal gain coefficient (cm-1) • In practice get both absorption and gain I(x) = I0 exp([g − a]x) • Gain is related directly to the population inversion g = g0 (N1 − N0 ) g0 = a constant for a given system • This seen in the Einstein B Coefficients • Thus laser needs gain medium to amplify signal Optical Resonator Cavity • In laser want to confine light: have it bounce back and forth • Then it will gain most energy from gain medium • Need several passes to obtain maximum energy from gain medium • Confine light between two mirrors (Resonator Cavity) Also called Fabry Perot Etalon • Have mirror (M1) at back end highly reflective • Front end (M2) not fully transparent • Place pumped medium between two mirrors: in a resonator • Needs very careful alignment of the mirrors (arc seconds) • Only small error and cavity will not resonate • Curved mirror will focus beam approximately at radius • However is the resonator stable? • Stability given by g parameters: g1 back mirror, g2 front mirror: L gi = 1 − ri • For two mirrors resonator stable if 0 < g1g2 < 1 • Unstable if g1g2 < 0 g1g2 > 1 • At the boundary (g1g2 = 0 or 1) marginally stable Stability of Different Resonators • If plot g1 vs g2 and 0 < g1g2 < 1 then get a stability plot • Now convert the g’s also into the mirror shapes Polarization and Lasers • Lasers often need output windows on gain medium in cavity • Output windows often produce polarized light • Normal windows lose light to reflection • Least loss for windows if light hits glass at Brewster Angle • Perpendicular polarization reflected • Parallel polarization transmitted with no loss (laser more efficient) • Called a Brewster Window & the Brewster Angle is −1⎛ n2 ⎞ θb = tan ⎜ ⎟ ⎝ n1 ⎠ n1 = index of refraction of air n2 = index of refraction of glass • Example: What is Brewster for Glass of n2 = 1.5 −1⎛ n2 ⎞ −1⎛1.5 ⎞ o θb = tan ⎜ ⎟ = tan ⎜ ⎟ = 56.6 ⎝ n1 ⎠ ⎝ 1 ⎠ Gain Medium & Laser Threshold • With good Gain Medium in optical cavity can get lasing but only if gain medium is excited enough • Low pumping levels: mostly spontaneous emission • At some pumping get population inversion in gain medium • Beyond inversion get Threshold pumping for lasing set by the losses in cavity • Very sensitive to laser cavity condition eg slight misalignment of mirrors threshold rises significantly • At threshold gain in one pass = losses in cavity Round Trip Power Gain • Within medium light intensity I gained in one pass I( L ) = I0 exp[(g −α )L] where g = small signal gain coefficient α = the absorption coefficient L = length of cavity • Thus calculate Round Trip Power Gain Gr • Each mirror has a reflectivity Ri R=1 for perfect reflection off a mirror I( 2L ) G = = R R exp[]()g −α 2L r I(0 ) 1 2 • At threshold Gr = 1 • Thus threshold gain required for lasing is 1 ⎛ 1 ⎞ gth = α + ln⎜ ⎟ 2L ⎝ R1R2 ⎠ • Some of the loss = laser emission Transition Line Shape • Distribution of energy levels creates width of emission • Usually Gaussian shape • Width broadened by many mechanisms • Most more important in gases • Doppler Broadening (movement of molecules) • Collision Broadening (atomic/molecular collisions) • Radiative Lifetime Broadening (finite lifetime of transitions) Axial Modes of Laser • Proper phase relationship only signal in phase in cavity • Thus cavity must be integer number of half wavelengths pλ L = 2 where p = integer • Results in frequency separation of peaks c Δν = 2L • Emission from the atomic transitions is a wider Gaussian • Result is axial modes imposed on Gaussian Transition spread Axial modes within Transition Gaussian • Each axial mode is Gaussian shape narrower than transition peak • eg for L=1 m Argon laser at 514 nm c 3.00x108 c 3.00x108 Δν = = = 150 MHz ν = = = 5.83x1014 Hz 2L 2 λ 5.14x10−7 • Thus emission much smaller than 0.0026% of frequency Much narrower than other light sources. Transverse Modes • Comes from microwave cavities • Some waves travel off axis, but within cavity • Result is Phase changes in repeating paths • These can change shape of output • Get local minimums (nulls) in the output beam shape • Reduce these by narrowing the beam • Called Transverse ElectroMagnetic, TEM Transverse Modes • Transverse ElectroMagnetic, TEM depend on cavity type • In cylindrical geometry two possible phase reversal orientations • horizontal (q) and vertical (r) • Show these as TEMqr • q and r give number of null's in output beam • Horizontal (q) gives number of vertical running nulls • Vertical (r) gives number of horizontal running nulls • Thus TEM12 has 2 vertical, 1 horizontal nulls • Special mode TEM01* or donut mode • comes from rapid switching from TEM01 to TEM10 Einstein Coefficients and Lasers • Recall the Einstein Emission/Absorption Coefficients • Consider again a 2 level system being pumped by light • A21 is the Einstein spontaneous emission Coefficient • B12 is the Einstein spontaneous absorption coefficient • B21 is the Einstein stimulated emission coefficient • At equilibrium the absorption from pumping equals the spontaneous and stimulated emission.
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